Simplifying Integration of ∫sec(x)dx using Substitution Technique: Step-by-Step Guide

∫sec(x)dx

To integrate ∫sec(x)dx, we can use a technique called integration by substitution

To integrate ∫sec(x)dx, we can use a technique called integration by substitution. Let’s start by considering the derivative of the function inside the integral, which is sec(x):

d/dx (tan(x)) = sec^2(x)

Notice that the derivative of tan(x) is sec^2(x), so we can rewrite sec(x)dx as (1/sec(x))(sec(x))dx, or simply (1/sec(x))(d/dx(tan(x))).

Let u = tan(x), so du/dx = sec^2(x). Rearranging this equation, we have du = sec^2(x)dx.

Substituting these values into the original integral, we have:

∫sec(x)dx = ∫(1/sec(x))(d/dx(tan(x))) dx

Now, we can substitute u and du into the integral:

∫sec(x)dx = ∫(1/u)du

This integral is much simpler and straightforward to solve. The integral of (1/u) with respect to u is ln|u| + C, where C is the constant of integration.

Therefore, the final solution is:

∫sec(x)dx = ln|tan(x)| + C

where C is the constant of integration.

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